Editing Map projection (section)

===Choosing a model for the shape of the body===
Projection construction is also affected by how the shape of the Earth or planetary body is approximated. In the following section on projection categories, the earth is taken as a [[sphere]] in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate [[ellipsoid]]. Whether spherical or ellipsoidal, the principles discussed hold without loss of generality.

Selecting a model for a shape of the Earth involves choosing between the advantages and disadvantages of a sphere versus an ellipsoid. Spherical models are useful for small-scale maps such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to justify using the more complicated ellipsoid. The ellipsoidal model is commonly used to construct [[topographic map]]s and for other large- and medium-scale maps that need to accurately depict the land surface. [[Latitude#Auxiliary latitudes|Auxiliary latitudes]] are often employed in projecting the ellipsoid.

{{anchor|geoid}}A third model is the [[geoid]], a more complex and accurate representation of Earth's shape coincident with what [[mean sea level]] would be if there were no winds, tides, or land. Compared to the best fitting ellipsoid, a geoidal model would change the characterization of important properties such as distance, [[#Conformal|conformality]] and [[#Equal-area|equivalence]]. Therefore, in geoidal projections that preserve such properties, the mapped [[Geographic coordinate system#Geographic latitude and longitude|graticule]] would deviate from a mapped ellipsoid's graticule. Normally the geoid is not used as an [[Earth model]] for projections, however, because Earth's shape is very regular, with the [[undulation of the geoid]] amounting to less than 100&nbsp;m from the ellipsoidal model out of the 6.3&nbsp;million&nbsp;m [[Earth radius]]. For irregular planetary bodies such as [[asteroids]], however, sometimes models analogous to the geoid are used to project maps from.<ref>
{{Cite journal
  | doi     = 10.1559/152304000783547957
  | title   = Equal Area Map Projection for Irregularly Shaped Objects
  | journal = Cartography and Geographic Information Science
  | volume  = 27
  | issue   = 2
  | page    = 91
  | year    = 2000
  | last1   = Cheng
  | first1  = Y.
  | last2   = Lorre
  | first2  = J. J.
  | s2cid = 128490229
 }}</ref><ref>
{{Cite journal
  | doi     = 10.1111/j.1541-0064.1998.tb01553.x
  | title   = Mapping Worlds with Irregular Shapes
  | journal = The Canadian Geographer
  | volume  = 42
  | page    = 61
  | year    = 1998
  | last1   = Stooke
  | first1  = P. J.
  }}</ref><ref>
{{cite journal
  | last1   = Shingareva
  | first1  = K.B.
  | last2   = Bugaevsky
  | first2  = L.M.
  | last3   = Nyrtsov
  | first3  = M.
  | title   = Mathematical Basis for Non-spherical Celestial Bodies Maps
  | journal = Journal of Geospatial Engineering
  | volume  = 2
  | issue   = 2
  | pages   = 45–50
  | year    = 2000
  | url     = http://www.lsgi.polyu.edu.hk/staff/ZL.Li/vol_2_2/06_nyrtsov.pdf
  }}</ref><ref>
{{cite journal
  | last1   = Nyrtsov
  | first1  = M.V.
  | date    = August 2003
  | title   = The Classification of Projections of Irregularly-shaped Celestial Bodies
  | journal = Proceedings of the 21st International Cartographic Conference (ICC)
  | pages   = 1158–1164
  | url     = http://icaci.org/files/documents/ICC_proceedings/ICC2003/Papers/141.pdf
  }}</ref><ref>
{{Cite book
  | doi     = 10.1007/978-1-4614-7762-4_6
  | chapter = CSNB Mapping Applied to Irregular Bodies
  | title   = Constant-Scale Natural Boundary Mapping to Reveal Global and Cosmic Processes
  | page    = 71
  | series  = SpringerBriefs in Astronomy
  | year    = 2013
  | last1   = Clark
  | first1  = P. E.
  | last2   = Clark
  | first2  = C. S.
  | isbn    = 978-1-4614-7761-7
  }}</ref> 

Other regular solids are sometimes used as generalizations for smaller bodies' geoidal equivalent. For example, [[Io (moon)|Io]] is better modeled by triaxial ellipsoid or prolated spheroid with small eccentricities. [[Haumea]]'s shape is a [[Jacobi ellipsoid]], with its major [[Axis of rotation|axis]] twice as long as its minor and with its middle axis one and half times as long as its minor.
See [[map projection of the triaxial ellipsoid]] for further information.